Algorithms for Programmers by Arndt J

By Arndt J

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The elements in the lower right triangle do not ‘wrap around’ anymore, they go to extra buckets. Note that bucket 31 does not appear, it is always zero. The equivalent table for a (cyclic) correlation is +-| 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 CHAPTER 2. CONVOLUTIONS 0: 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 15 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 14 15 0 1 2 3 4 5 6 7 8 9 10 11 12 13 13 14 15 0 1 2 3 4 5 6 7 8 9 10 11 12 12 13 14 15 0 1 2 3 4 5 6 7 8 9 10 11 11 12 13 14 15 0 1 2 3 4 5 6 7 8 9 10 40 10 11 12 13 14 15 0 1 2 3 4 5 6 7 8 9 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 8 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 7 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 6 7 8 9 10 11 12 13 14 15 0 1 2 3 4 5 5 6 7 8 9 10 11 12 13 14 15 0 1 2 3 4 4 5 6 7 8 9 10 11 12 13 14 15 0 1 2 3 3 4 5 6 7 8 9 10 11 12 13 14 15 0 1 2 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 while the acyclic counterpart is: +-| 0: 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 31 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 30 31 0 1 2 3 4 5 6 7 8 9 10 11 12 13 29 30 31 0 1 2 3 4 5 6 7 8 9 10 11 12 28 29 30 31 0 1 2 3 4 5 6 7 8 9 10 11 27 28 29 30 31 0 1 2 3 4 5 6 7 8 9 10 26 27 28 29 30 31 0 1 2 3 4 5 6 7 8 9 25 26 27 28 29 30 31 0 1 2 3 4 5 6 7 8 24 25 26 27 28 29 30 31 0 1 2 3 4 5 6 7 23 24 25 26 27 28 29 30 31 0 1 2 3 4 5 6 22 23 24 25 26 27 28 29 30 31 0 1 2 3 4 5 21 22 23 24 25 26 27 28 29 30 31 0 1 2 3 4 20 21 22 23 24 25 26 27 28 29 30 31 0 1 2 3 19 20 21 22 23 24 25 26 27 28 29 30 31 0 1 2 18 19 20 21 22 23 24 25 26 27 28 29 30 31 0 1 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 0 Note that bucket 16 does not appear, it is always zero.

THE FOURIER TRANSFORM 34 A level of abstraction for the array indices is of great use: When the print statements in the generator emit some function of the index instead of its plain value it is easy to generate modified versions of the code for permuted input. g. the revbin-permuted) index k. 75/pt The generated codes can be of great use when one wants to spot parts of the original code that need further optimization. Especially repeated trigonometric values and unused symmetries tend to be apparent in the unrolled code.

C − 1 and y = 0, 1, 2, . . , R − 1) be a 2-dimensional array of data7 . 77) k=0 h=0 For a m-dimensional array ax (x = (x1 , x2 , x3 , . . , xm ), xi ∈ 0, 1, 2, . . , Si ) the m-dimensional Fourier transform ck (k = (k1 , k2 , k3 , . . , km ), ki ∈ 0, 1, 2, . . k ... x1 =0 x2 =0 where z = e± 2 π i/n , n = S1 S2 . . k where S = (S1 − 1, S2 − 1, . . 79) x=0 The inverse transform is again the one with the minus in the exponent of z. 80) x=0 which shows that the 2-dimensional FT can be accomplished by using 1-dimensional FTs to transform first the rows and then the columns8 .

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